A Model of Unconventional Monetary Policy

Mark Gertler and Peter Karadi

NYU

April 2009

Abstract

This paper develops a quantitative monetary DSGE model thatallows for financial intermediaries that face endogenous balance sheetconstraints. We use the model to simulate a crisis that has some basicfeatures of the current economic downturn. We then use the model toquantitatively assess the effect of direct central bank intermediation ofprivate lending, which is the essence of the unconventional monetarypolicy that the Federal Reserve has developed to combat the subprimecrisis. We show numerically how central bank credit policy might helpmoderate the simulated crisis. We then compute the optimal degreeof central bank credit intervention in this scenario and also computethe welfare gains.

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1 Introduction

Over most of the post-war the Federal Reserve has conducted monetary pol-icy by manipulating the Federal Funds rate in order to affect market interestrates.. It has avoided lending directly in private credit markets, other thanto supply discount window loans to commercial banks. Even then, it limiteddiscount window activity to loans secured by government Treasury BillsSince the onset of the subprime crisis in August 2007, the situation has

changed dramatically. To address the deterioration in both financial and realactivity, the Fed has directly injected credit into private markets. It beganin the fall of 2007 by expanding the range of eligible collateral for discountwindow loans to include agency debt and high grade private debt,. It did soin conjunction with extending the maturity of these types of loans and withextending eligibility to investment banks. Since that time, the Fed has set upa myriad of lending facilities. For example, it has provided backstop fundingto help revive the commercial paper market. It has also intervened heavily inmortgage markets by directly purchasing agency debt and mortgage-backedsecurities. (That mortgage rates have fallen significantly since the onsetof this program is perhaps the strongest evidence of a payoff to the Fed’soverall new strategy.) Plans are in the works to extend central bank creditto securitized consumer loans.The Fed’s balance sheet provides the most concrete measure of its credit

market intervention:.Since August 2007 the quantity of private assets it hasheld has increased from virtually nothing to nearly a trillion and half. Ef-fectively, over this period the Fed has attempted to offset the decline of aconsiderable fraction of private financial intermediation by expanding centralbank intermediation. Now that the short term interest rate is at the zerolower bound, the Fed is unable to stimulate.the economy using conventionalmeans. For time being, it must rely exclusively on its new unconventionalbalance sheet operations.At the same time, operational models of monetary policy have not kept

pace with the dramatic changes in actual practice. There is of a course alengthy contemporary literature on quantitative modeling of conventionalmonetary policy, beginning with Christiano, Eichenbaum and Evans (2005)and Smets and Wouters (2007.) The baseline versions of these models, how-ever, assume frictionless financial markets. They are thus unable to capturefinancial market disruptions that could motivate the kind of central bankinterventions in loan markets that are currently in play. Similarly, models

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which do incorporate financial market frictions, such as Bernanke, Gertlerand Gilchrist (1999) or Christiano, Motto and Rostagno (2005), have not yetexplicitly considered direct central bank intermediation as a tool of monetarypolicy. Work that has tried to capture this phenomenon has been mainlyqualitative as opposed to quantitative (e.g., Kiyotaki and Moore (2008),Adrian and Shin (2008).). Accordingly, the objective of this paper is totry to fill in this gap in the literature: The specific goal is develop a quan-titative macroeconomic model where it is possible to analyze the effects ofunconventional monetary policy in the same general manner that existingframeworks are able to study conventional monetary policy.To be clear, we do not attempt to explicitly model the sub-prime cri-

sis. However, we do try to capture the key elements relevant to analyzingthe Fed’s credit market interventions. In particular, the current crisis hasfeatured a sharp deterioration in the balance sheets of many key financialintermediaries. As many observers argue, the deterioration in the financialpositions of these institutions has had the effect of disrupting the flow offunds between lenders and borrowers. Symptomatic of this disruption hasbeen a sharp rise in various key credit spreads as well as a significant tight-ening of lending standards This tightening of credit, in turn, has raised thecost of borrowing and thus enhanced the downturn. The story does not endhere: The contraction of the real economy has reduced asset values through-out, further weakening intermediary balance sheets, and so on. It is in thiskind of climate, that the central bank has embarked on its direct lendingprograms.To capture this kind of scenario, accordingly we incorporate financial in-

termediaries within an otherwise standard macroeconomic framework. Tomotivate why the condition of intermediary balance sheets influences theoverall flow of credit, we introduce a simple agency problem between inter-mediaries and their respective depositors. The agency problem introducesendogenous constraints on intermediary leverage ratios, which have the ef-fect of tieing overall credit flows to the equity capital in the intermediarysector. As in the current crisis, a deterioration of intermediary capital willdisrupt lending and borrowing in a way that raises credit costs.To capture unconventional monetary policy in this environment, we allow

the central bank to act as intermediary by borrowing funds from savers andthen lending them to investors. Unlike, private intermediaries, the centralbank does not face constraints on its leverage ratio. (There is no agencyproblem between the central bank and its creditors because it can commit

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to always honoring its debt.) Thus, in a period of financial distress that hasdisrupted private intermediation, the central bank can intervene to supportcredit flows. On the other hand, we allow for the fact that, everything elseequal, public intermediation is likely to be less efficient the private interme-diation. When we use the model to evaluate these credit interventions, wetake into account this trade-off,Section 2 presents the baseline model. The framework is closely related

to financial accelerator model developed by Bernanke, Gertler and Gilchrist(BGG, 1999). That approach emphasized how balance sheet constraintscould limit the ability of non-financial firms to obtain investment funds.Firms effectively borrowed directly from households and financial interme-diaries were simply a veil. Here financial intermediaries, as we discussed,financial intermediaries are may be subject to endogenously determined bal-ance sheet constraints. In addition, we allow for the central bank to lenddirectly private credit markets.Another difference from BGG is that, we use as a baseline framework the

conventional monetary business cycle framework developed by Christiano,Eichenbaum and Evans (CEE, 2005), Smets and Wouters (SW, 2007) andothers. We adopt this approach because this framework has proven to havereasonable empirical properties. Here we use it to study not only conventionalinterest policy but also unconventional credit market interventions by thecentral bank.Section 3 presents a quantitative analysis of the model. We illustrate how

financial factors may amplify and propagate some conventional disturbances.We also consider a disturbance to the underlying quality of intermediaryassets and then show how this kinds of disturbance could create a contractionin real activity that mirrors some of the basic features of the current crisis. Wethen illustrate the extent to which central credit interventions could moderatethe downturn.In section 4, we undertake a normative analysis of credit policy. We first

solve for the optimal central credit intervention in crisis scenario considered insection 3. We do so under different assumptions about the efficiency costs ofcentral bank intermediation. We then compute for each case the net welfaregains from the optimal credit market intervention. We find that so long asthe efficiency costs are quite modest, the gains may be quite significant. Aswe discuss in the concluding remarks in section 5, this finding suggests a wayto think about the central bank’s choice between direct credit interventionsversus alternatives such as equity injections to private intermediaries.

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2 The Baseline Model

The core framework is the monetary DSGE model with nominal rigidities de-veloped by CEE and SW. To this we add financial intermediaries that trans-fer funds between households and non-financial firms. An agency problemconstrains the ability of financial intermediaries to obtain from households.Another new feature is a disturbance to the quality of capital. Absent fi-nancial frictions, this shock introduces only a modest decline in output, asthe economy works to replenish the effective capital stock. With frictions inthe intermediation process, however, the shock creates a significant capitalloss in the financial sector, which in turn induces tightening of credit and asignificant downturn. As we show, it is in this kind of environment that theis a potential role for central bank credit interventions.There are five types of agents in the model: Households, financial inter-

mediaries, non-financial goods producers, capital producers, and monopolis-tically competitive retailers. The latter are in the model only to introducenominal price rigidities. In addition, there is a central bank that conductsboth conventional and unconventional monetary policy. Without financialintermediaries the model is isomorphic to CEE and SW. As we show, thoughaddition of financial intermediaries adds only a modest degree of complexity.It has, however, a substantially on model dynamics and associated policyimplications..We now proceed to characterize the basic ingredients of the model.

2.1 Households

There a continuum of identical households of measure unity. Each householdconsumes, saves and supplies labor. Households save by lending funds tocompetitive financial intermediaries and possibly also by lending funds tothe government.Within each household there are two types of members: workers and

bankers. Workers supply labor and return the wages they earn to the house-hold. Each banker manages a financial intermediary and similarly transfersany earnings back to household. The household thus effectively owns theintermediaries that its bankers manage. The deposits it holds, however, arein intermediaries that is does not own. Finally, within the family there isperfect consumption insurance. As we make clear in the next section, thissimple form of heterogeneity within the family allows us to introduce finan-

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cial intermediation in a meaningful way within an otherwise representativeagent framework.At any moment in time the fraction 1− f of the household members are

workers and the fraction f are bankers. Over time an individual can switchbetween the two occupations. In particular, a banker this period stays bankernext period with probability θ, which is independent history (i.e., of how longthe person has been a banker.) The average survival time for a banker in anygiven period is thus 1

1−θ . As will become clear, we introduce a finite horizonfor bankers to insure that over time they do not reach the point where theycan fund all investment from their own capital. Thus every period (1− θ)fbankers exit and become workers. A similar number of workers randomlybecome bankers, keeping the relative proportion of each type fixed. Bankerswho exit give their retained earnings to their respective household. Thehousehold, though, provides its new bankers with some start up funds, as wedescribe in the next sub-section.Let Ct be consumption and Lt family labor supply. Then households

preferences are given by

maxEt

∞Xi=0

βi[ln(Ct+i − hCt+i−1)−χ

1 + ϕL1+ϕt+i ] (1)

with 0 < β < 1, 0 < h < 1 and χ, ϕ > 0. As in CEE and SW we allow forhabit formation to capture consumption dynamics. As in Woodford (2003)we consider the limit of the economy as it become cashless, and thus ignorethe convenience yield to household’s from real money balances.Both intermediary deposits and government debt are one period real

bonds that pay the gross real return Rt from t−1 to t. In the equilibrium weconsider, the instruments are both riskless and are thus perfect substitutes.Thus, we impose this condition from the outset. Thus let Bt be the totalquantity of short term debt the household acquires, Wt, be the real wage, Πt

net payouts to the household from ownership of both non-financial and fi-nancial firms and, Tt lump sum taxes. Then the household budget constraintis given by

Ct =WtLt +Πt + Tt +RtBt −Bt+1 (2)

Note that Πt is net the transfer the household gives to its members that enterbanking at t.

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Let t denote the marginal utility of consumption. Then the household’sfirst order conditions for labor supply and consumption/saving are standard:

tWt = χLϕt (3)

with

t = Ct − hCt−1)−1 − βhEt(Ct+1 − hCt)

−1

and

EtβΛt,t+1Rt+1 = 1 (4)

with

Λt,t+1 ≡λt+1λt

2.2 Financial Intermediaries

Financial intermediaries lend funds obtained from households to non-financialfirms. LetNjt be the amount of wealth - or net worth - that a banker/intermediaryj has at the end of period t; Bjt the deposits the intermediary obtains fromhouseholds, Sjt the quantity of financial claims on non-financial firms that theintermediary holds and Qt the relative price of each claim. The intermediarybalance sheet is then given by

QtSjt = Njt +Bjt (5)

For the time being, we ignore the possibility of the central bank supplyingfunds to the intermediary.As we noted earlier, household deposits with the intermediary at time

t, pay the non-contingent real gross return Rt+1 at t + 1. Thus Bjt may bethought of as the intermediary’s debt and Njt as its equity capital. Interme-diary assets earn the stochastic return Rkt+1 over this period. Both Rkt+1

and Rt+1 will be determined endogenously.Over time, the banker’s equity capital evolves as the difference between

earnings on assets and interest payments on liabilities:

Njt+1 = Rkt+1QtSjt −Rt+1Bjt (6)

= (Rkt+1 −Rt+1)QtSjt +Rt+1Njt (7)

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Any growth in equity above the riskless return depends on the premiumRkt+1 − Rt+1 the banker earns on his assets, as well as his total quantity ofassets, QtSjt .Let βΛt,t+i be the stochastic discount the the banker at t applies to earn-

ings at t+ i. Since the banker will not fund assets with a discounted returnless than the discounted cost of borrowing, for the intermediary to operatethe following inequality must apply:

EtβΛt,t+1+i(Rkt+1+i −Rt+1+i) ≥ 0 ∀ i ≥ 0

With perfect capital markets, the relation always holds with equality: Therisk-adjusted premium is zero. With imperfect capital markets, however, thepremiummay be positive due to limits on the intermediary’s ability to obtainfunds.So long as the intermediary can earn a risk adjusted return that is greater

than or equal to the return the household can earn on its deposits, it pays forthe banker to keep building assets until exiting the industry. Accordingly,the banker’s objective is to maximize expected terminal wealth, given by

Vjt = maxEt

Xi

(1− θ)θiβiΛt,t+i(Njt+1+i) (8)

= maxEt

Xi

(1− θ)θiβiΛt,t+i[(Rkt+1+i −Rt+1+i)Qt+iSjt+i +Rt+1+iNjt+i]

To the extent the discounted risk adjusted premium in any period, βiΛt,t+i[(Rkt+1+i−Rt+1+i), is positive, the intermediary will want to expand its assets indefi-nitely by borrowing additional funds from households. To motivate a limiton its ability to do so, we introduce the following moral hazard/costly en-forcement problem: At the beginning of the period the banker can choose todivert the fraction λ of available funds from the project and instead transferthem back to the household of which he or she is a member. The cost to thebanker is that the depositors can force the intermediary into bankruptcy andrecover the remaining fraction 1− λ of assets.. However, it is too costly forthe depositors recover the fraction of funds that the banker diverted.Accordingly for lenders to be willing to supply funds to the banker, the

following incentive constraint must be satisfied:

Vjt ≥ λQtSjt (9)

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The left side is what the banker would lose by diverting a fraction of assets.The right side is the gain from doing so.We can express Vjt as follows:

Vjt = vt ·QtSjt + ηtNjt (10)

with

vt = Et(1− θ)βΛt,t+1(Rkt+1 −Rt+1) + βΛt,t+1θxt,t+1vt+1 (11)

ηt = Et(1− θ) + βΛt,t+1θzt,t+1ηt+1

where xt,t+i ≡ Qt+iSjt+i/QtSjt, is the gross growth rate in assets between tand t + i, and zt,t+i ≡ Njt+i/Njt is the gross growth rate of net worth. Thevariable vt has the interpretation of the expected discounted marginal gainto the banker of expanding assets QtSjt by a unit, holding net worth Njt con-stant, and while ηt is the expected discounted value of having another unityof Njt,holding Sjt constant. With frictionless competitive capital markets,intermediaries will expand borrowing to the point where rates of return willadjust to ensure vt is zero. The agency problem we have introduced, however,may place limits on the arbitrage. In particular, as we next show, when theincentive constraints is binding, the intermediary’s assets are constrained byits equity capital.Note first that we can express the incentive constraints as

ηtNjt + vtQtSjt ≥ λQtSjt (12)

If this constraint binds, then the assets the banker can acquire will dependpositively on his/her equity capital:

QtSjt =ηt

λ− vtNjt (13)

= φtNjt

where φt ratio of privately intermediated assets to equity, which we willrefer to as the (private) leverage ratio.. Holding constant Njt, expandingSjt raises the bankers’ incentive to divert funds. The constraint (13) limitsthe intermediaries leverage ratio to the point where the banker’s incentiveto cheat is exactly balanced by the cost. In this respect the agency problem

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leads to an endogenous capital constraint on intermediary’s ability to acquireassets.Given Njt > 0, the constraint binds only if 0 < vt < λ. In this instance,

it is profitable for the banker to expand assets (since vt > 0). Note that inthis circumstance the leverage ratio that depositors will tolerate is increasingin vt. The larger is vt, the greater is the cost to the banker from being forcedinto bankruptcy. If vt increases above λ, the incentive constraint does notbind: the franchise value of the intermediary always exceed the gain fromdiverting funds. In the equilibrium we construct below, under reasonableparameter values the constraint always binds.We can now express the evolution of the banker’s net worth as

Njt+1 = [(Rkt+1 −Rt+1)φt +Rt+1]Njt (14)

Note that the sensitivity of Njt+1 to the ex post realization of the excessreturn Rkt+1 − Rt+1 is increasing in the leverage ratio φt. In addition, itfollows that

zt,t+1 = Njt+1/Njt = (Rkt+1 −Rt+1)φt +Rt+1

xt,t+1 = Qt+1Sjt+2/QtSt+1 = (φt+1/φt)(Njt+1/Nt) = (φt+1/φt)zt,t+1

Importantly, all the components of φt do not depend on firm-specificfactors. Thus to determine total intermediary demand for assets we can sumacross individual demands to obtain:

QtSIt = φtNt (15)

where SIt reflects the aggregate quantity of intermediary assets and Nt de-notes aggregate intermediary capital. In the general equilibrium of our model,variation in Nt, will induce fluctuations in overall asset demand by interme-diaries. Indeed, a crisis will feature a sharp contraction in Nt.We can derive an equation of motion for Nt, by first recognizing that it

is the sum of the net worth of existing banker/intermediaries, Net, and thenet worth of entering (or "new") bankers, Nnt.

Nt = Net +Nnt (16)

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Since the fraction θ of bankers at t− 1 survive until t, Net is given by

Net = θ[(Rkt −Rt)φt +Rt]Nt−1 (17)

Observe that the main source of variation in Net will be fluctuations in theex post return on assets Rkt. Further, the impact on Net is increasing in theleverage ratio φt.As we noted earlier, newly entering bankers receive "start up" funds from

their respective households. We suppose that the startup money the house-hold gives its new banker a transfer equal to a small fraction of the value ofassets that exiting bankers had intermediated in their final operating period.The rough idea is that how much the household feels that is new bankersneed to start, depends on the scale of the assets that the exiting bankershave been intermediating. Given that the exit probability is i.i.d., the totalfinal period assets of exiting bankers at t is (1 − θ)QtSt−1. Accordingly weassume that each period the household transfers the fraction ξ/(1−θ) of thisvalue to its entering bankers. Accordingly, in the aggregate,

Nnt = ξQtSt−1 (18)

Combining (17) and (18) yields the following equation of motion for Nt.

Nt = θ[(Rkt −Rt)φt +R]Nt−1 + ξQtSt−1

Observe that ξ helps pin down the steady state leverage ratio QS/N . Indeed,in the next section we calibrate ξ to match this evidence. The resulting value,as we show, is quite small.

2.3 Credit Policy

In the previous section we characterized how the total value of privately inter-mediated assets, QtSpt, is determined. We now suppose that the central bankis willing to facilitate lending. Let QtSgt be the value of assets intermediatedvia government assistance and let QtSt be the total value of intermediatedassets: i.e.,

QtSt = QtSpt +QtSgt (19)

To conduct credit policy, the central bank issues government debt tohouseholds that pays the riskless rate Rt+1 and then lends the funds to non-financial firms at the market lending rate Rkt+1.We suppose that government

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intermediation involves efficiency costs: In particular, the central bank creditinvolves an efficiency cost of τ per unit supplied. This deadweight loss couldreflect the costs of raising funds via government debt. It might also reflectcosts to the central bank of identifying preferred private sector investments.On the other hand, the government always honors its debt: Thus, unlike thecase with private financial institutions ,is no agency conflict than inhibits thegovernment from obtaining funds from households. Put differently, unlikeprivate financial intermediation, government intermediation is not balancesheet constrained.An equivalent formulation of credit policy involves having the central

bank channel funds to non-financial borrowers via private financial interme-diaries, as has occurred in practice. Here we assume that the government hasan advantage over private households in enforcing in enforcing payment ofdebts by private intermediaries. In particular, it is not possible for an inter-mediary to walk away from a financial obligation to the federal government,the same way it can a private entity. Unlike private creditors, the federalgovernment has various means to track down and recover debts. It followsthat In this instance, the balance sheets constraints that limit intermediariesability to obtain private credit do not constrain their ability to obtain cen-tral bank credit. Accordingly, in this scenario, after obtaining funds fromhouseholds at the rate Rt+1, the central bank lends freely to private finan-cial intermediaries at the rate Rkt, which in turn lend to non-financial firmsat the same rate. Private intermediaries earn zero profits on this activity:the liabilities to the central bank perfectly offset the value of the claims onnon-financial firms, implying that there is no effect on intermediary balancesheets. The behavior of the model is thus exactly same as if the centralbank directly lends to non-financial firms. Note that in this instance, theefficiency cost τ is interpretable as the cost of publicly channeling funds toprivate intermediaries as opposed to directly to non-financial firms.Accordingly, suppose the central bank is willing to fund the fraction ψt

of intermediated assets: i.e.,

QtSgt = ψtQtSt (20)

It issues amount of government bonds Bgt, equal to ψtQtSt to funds thisactivity. It’s net earnings from intermediation in any period t thus equal(Rkt+1−Rt+1)Bgt. These net earnings provide a source of government revenueand must be accounted for in the budget constraint, as we discuss later.

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Since privately intermediated funds are constrained by intermediary networth, we can rewrite equation (19) to obtain

QtSt = φtNt + ψtQtSt

= φctNt

where φt is the leverage ratio for privately intermediated funds (see equations(13) and (15), and where φct is the leverage ratio for total intermediatedfunds, public as well as well private.

φct =1

1− ψt

φt

Observe that φct depends positively on the intensity of credit policy, as mea-sured by ψt. Later how describe how the central might choose ψt to combata financial crisis.

2.4 Intermediate Goods Firms

We next turn to the production and investment side of the model economy.Competitive non-financial firms produce intermediate goods that are even-tually sold to retail firms. The timing is as follows: At the end of period t,an intermediate goods producer acquires capital Kt+1 for use in productionin the subsequent period. After production in period t+ 1, the firm has theoption of selling to capital on the open market. There are no adjustmentcosts at the firm level. Thus, the firm’s capital choice problem is alwaysstatic, as we discuss below.The firm finances its capital acquisition each period by obtaining funds

from intermediaries. To acquire the funds to buy capital, the firm issues Stclaims equal to the number of units of capital acquired Kt+1 and prices eachclaim at the price of a unit of capital Qt. That is, QtKt+1 is the value ofcapital acquired and QtSt is the value of claims against this capital. Thenby arbitrage:

QtKt+1 = QtSt (21)

We assume that there are no frictions in the process of non-financialfirms obtaining funding from intermediaries. The intermediary has perfectinformation about the firm and has no problem enforcing payoffs. This con-trasts with the process of the intermediary obtaining funding from house-holds. Thus, within our model, only intermediaries face capital constraints

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on obtaining funds. These constraints, however, affect the supply of fundsavailable to non-financial firms and hence the required rate of return on cap-ital these firms must pay. Conditional on this required return, however, thefinancing process is frictionless for non-financial firms.At time t+1 the firm produces output Yt+1, using capital and labor Lt+1,

and by varying the utilization rate of capital, Ut+1. Let At+1 denote totalfactor productivity. Then production is given by:

Yt+1 = At+1(Ut+1Kt+1)αL1−αt+1 (22)

What the firm earns in t + 1 is the value of output plus the value of itscapital stock left over net financing and labor costs. Let Pmt be the price ofintermediate goods output. Then from the vantage of period t, where thefirm makes its’ capital decision for t+ 1, its objective is given by:

maxEtβΛt,t+1[Pmt+1Yt+1+(Qt+1−δ(Ut))ξt+1Kt+1−Rt+1QtKt+1−Wt+1Lt+1]

where as before βΛt,t+1 is the firm’s stochastic discount factor, δ(Ut) is thecapital depreciation rate, which is increasing and convex in Ut,Wt+1 is thereal wage, and Rt+1 is the state-contingent required return on capital. Inaddition ψt+1 is an exogenous factor that affects the effective quantity ofcapital. That is, after production in t+1, the number of units of capital leftover is (1− δ(Ut))ψt+1Kt+1. Assuming that capital that replacement price ofcapital that has depreciated is unity, then the value of the capital stock thatis left over is given by (Qt+1− δ(Ut))ψt+1Kt+1. Finally, ψt+1 may be thoughtof as a measure of the quality of the existing.Maximizing with respect to Kt+1 yields

EtβΛt,t+1Rt+1 = EtβΛt,t+1

Pmt+1αYt+1Kt+1

+ (Qt+1 − δ(Ut+1))ξt+1

Qt (23)

At an interior optimum, the discounted cost of capital must equal the dis-counted return.At t+1, the firm chooses the utilization rate and labor demand as follows:

Pmt+1αYt+1Ut+1

= δ0(Ut+1) (24)

Pmt+1αYt+1Lt+1

=Wt+1 (25)

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2.5 Capital Producing Firms

At the end of period t, competitive capital producing firms buy capital fromintermediate goods producing firms and then repair depreciated capital andbuilds new capital. They then sell both the new and re-furbished capital. Aswe noted earlier, the cost of replacing worn out capital is unity. The value ofa unit of new capital is Qt, as is the value of a unit of re-furbished capital.If It is total investment by a capital producing firm, then the firms profits att are given by.

QtKt+1 − (Qt − δ(Ut))ξtKt − It

Let Int ≡ It − δ(Ut))ξtKt, be the firm’s objective. Then we can express ismaximization problem, as to choose Int, Kt+1 and Kt to solve

maxQt(Kt+1 − ξtKt)− Int (26)

subject to

Kt+1 − ξtKt = Int − Sµ

Int + IssInt−1 + Iss

¶(Int + Iss) (27)

where Iss is steady state investment, which consists only of replacement in-vestment. As in CEE, we allow for flow adjustment costs of investment, butrestrict these costs to modify the net investment flow.The first order condition for investment gives the follow ”Q” relation for

net investment:

Qt = [1−Sµ

IantIant−1

¶−S 0

µIantIant−1

¶µIantIant−1

¶]−1[1−βEtΛt,t+1Qt+1S

0µIant+1Iant

¶µIant+1Iant

¶2]

(28)where Iant ≡ Int + Iss may be thought of as "adjusted" net investment.

2.6 Retail Firms

Final output Yt is a CES composite of a continuum of mass unity of differen-tiated retail firms, that use intermediate output as the sole input. The finaloutput composite is given by

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Yt = [

Z 1

0

Yftε−1ε− df ]

εε−1 (29)

where Yft is output by retailer f . From cost minimization by users of finaloutput:

Yft = (Pft

Pt)−εYt (30)

Pt = [

Z 1

0

Pft

11−ε

df ]1−ε (31)

Retailers simply re-package intermediate output. It takes one unit ofintermediate output to make a unit of retail output. The marginal costis thus the relative intermediate output price Pmt. We introduce nominalrigidities following CEE. In particular, each firm period a firm is able tofreely adjusts price with probability 1 − ς. In between these periods, thefirm is able to index its price to the lagged rate of inflation. The retailerspricing problem then is to choose the optimal reset price P ∗t to solve

max∞Xi=0

θiβiΛt,t+i[P ∗tPt+i

iYk=0

(1 + πt+i−1)− Pmt+i]Yft (32)

where πt is the rate of inflation from t − i to t. The first order necessaryconditions are given by:

∞Xi=0

θiβiΛt,t+i[P ∗tPt+i

iYk=0

(1 + πt+i−1)− μPmt+i]Yft = 0 (33)

withμ =

1

1− 1/εFrom the law of large numbers, the following relation for the evolution of theprice level emerges.

Pt = [(1− θ)(P ∗t )1

1−ε + θ(Πt−1Pt−1)1

1−ε ]1−ε (34)

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2.7 Resource Constraint and Government Policy:

Output is divided between consumption, investment, government consump-tion, Gt and expenditures on government intermediation, τψtQtKt+1. Wesuppose further that government expenditures are exogenously fixed at thelevel G. The economy-wide resource constraint is thus given by

Yt = Ct + It +G+ τψtQtKt+1 (35)

Government expenditures, further, are financed by lump sum taxes andgovernment intermediation:

G = Tt + (Rkt −Rt)Bgt−1

where government bonds, Bgt−1, finance total government intermediated as-sets, Qtψt−1St−1.We suppose monetary policy is characterized by a simple Taylor rule with

interest-rate smoothing. Let it be the net nominal interest rate, i the steadystate nominal rate, and Y ∗t the natural (flexible price equilibrium) lever ofoutput. Then:

it = (1− ρ)[i+ ιππt + ιy(log Y∗t − log Yt) + ρit−1 + t (36)

where the smoothing parameter ρ lies between zero and unity, and where t isan exogenous shock to monetary policy, and where the link between nominaland real interest rates is given by the following Fisher relation

1 + it = Rt+1Pt+1

Pt(37)

Finally, we also introduce a feedback role for credit policy. We sup-pose that the central bank injects credit in response to movements in creditspreads, according to the following feedback rule:

ψt = ψ + ν[(Rkt+1 −Rt+1)− (Rk −R)] (38)

where ψ is the steady state fraction of publicly intermediated assets andRk − R is the steady state premium. In addition, the feedback parameterexceeds unity. According to this rule, the central bank expands credit as thespread increase relative to its steady state value.This completes the description of the model.

17

3 Model Analysis

3.1 Calibration

Table 1 list the choice of parameter values for our baseline model. Overallthere are eighteen parameters. Fifteen are conventional. Three (λ, ξ, θ) arespecific to our model.We begin with the conventional parameters. For the discount factor β, the

depreciation rate δ, the capital share α, the elasticity of substitution betweengoods, ε, and the government expenditure share, we choose conventionalvalues. Also, we normalize the steady state utilization rate u at unity. We useestimates from Justinano, Primiceri and Tambalotti (2009) to obtain valuesfor the following preference and technology parameters: the habit parameterh, the elasticity of marginal depreciation with respect to the utilization rate,ζ, and the inverse elasticity of net investment to the price of capital ηi. Wepick the relative utility weight on labor χ to fix hours worked at one third ofavailable time. Finally, we choose a Frisch elasticity of labor supply equal to3. Because, unlike most of the existing quantitative models. we do not allowfor wage rigidity, we choose a relatively high labor supply elasticity, thoughone that is not inconsistent with the evidence if we interpret this elasticityas applying to the extensive margin.We the price rigidity parameter, γ, to have prices fixed on average for

a year. We choose a high value to compensate for not including real pricerigidities in the model. Our parametrization leads to a slope coefficient inthe Phillips curve that is consistent with the evidence. In addition, we setindexing parameter γp at .5, following CEE. Finally, the feedback coefficientsin the monetary policy rule, κπ and κy obey a conventional Taylor rule, witha smoothing parameter ρ. This rule is consistent with the evidence for post-1984.Our choice of the financial sector parameters - the fraction of capital that

can be divertedλ, the proportional transfer to entering bankers ξ, and thesurvival probability θ - is meant to be suggestive. We pick these parame-ters to hit the following three targets: a steady state interest rate spreadof one hundred basis points; a steady state leverage ratio of four; and anaverage horizon of bankers of a decade. We base the steady state target forthe spread on the pre-2007 spreads between mortgage rates and governmentbonds and between BAA corporate vs. government bonds. The choice of theleverage ratio is a rough guess of a reasonable economy-wide mortgage. For

18

the mortgage sector, which was about one third of total assets in 2007, thisratio was between twenty and thirty to one. It was obviously much smallerin other sectors.

3.2 Experiments

We begin with several experiments designed to illustrate how the model be-haves. We then consider a "crisis" experiment that mimics some of the basicfeatures of the current downturn. We then consider the role of central bankcredit policy in moderating the crisis.Figure 1 shows the response of the model economy to three disturbances:

a technology shock, a monetary shock, and shock to intermediary net worth.In each case, the direction of the shock is set to produce a downturn. Thefigure then shows the responses of three key variables: output, investmentand the premium. In each case the solid shows the response of the baselinemodel. The dotted line gives the response of the same model, but with thefinancial frictions removed.The technology shock is a negative one percent innovation in TFP, with a

quarterly autoregressive factor of 0.95. The intermediary balance mechanismproduces a modest amplification of the decline in output the baseline modelrelative to the conventional DSGE model. The amplification is mainly theproduct of substantially enhanced decline investment:on the order of fiftypercent relative to the frictionless model. The enhanced response of invest-ment in the balance model is a product of the rise in the premium, plottedin the last panel on the right. The unanticipated decline in investment re-duces asset prices, which produces a deterioration an intermediary balancesheets, pushing up the premium. The increase in the cost of capital, furtherreduces capital demand by non-financial firms, which enhances the downturnin investment and asset prices. In the conventional model without financialfrictions, of course, the premium is fixed at zero.The monetary shock is an unanticipated twenty-five basis point increase in

the short term interest rate. The effect on the short term interest rate persistsdue to interest rate smoothing by the central bank. Financial frictions leadto greater amplification relative to the case of the technology shock. Thisenhanced amplification is due to he fact that, everything else equal, themonetary policy shock has a relatively large effect on investment and assetprices. The latter triggers the financial accelerator mechanism.To illustrate how at the core of the amplification mechanism in the first

19

two experiments is procyclical variation in intermediary balance sheets, weconsider a redistribution of wealth from intermediaries to households. Inparticular, we suppose that intermediary net worth declines by one percentand is transferred to households. In the model with no financial frictions, thisredistribution has no effect (it is just a transfer of wealth within the family.)The decline in intermediary in our baseline model, however, produces a risein the premium, leading to a subsequent decline in output and investment.We now turn to the crisis experiment. The initiating disturbance is a

decline in capital quality. What we are trying to capture, is a shock to thequality of intermediary assets that produce a enhanced decline in the capitalof these institutions, due to their high degree of leverage. In this rough way,we capture the broad dynamics of the sub-prime crises. We fix the size ofthe shock so that the downturn is of broadly similar magnitude to the onewe have recently experienced.We first consider the disturbance to the economy without credit policy

and then illustrate the effects of credit policy. The initiating shock is a fivepercent decline in capital quality, with a quarterly autoregressive factor of0.66. Absent any changes in investment, the shock produces a roughly tenpercent decline in effective capital stock over a two year period. The lossin value of the housing stock relative to the total capital stock was in thisneighborhood.In the model without financial frictions, then shock produces only a mod-

est decline in output. Output falls a bit initially due to the reduced effectivecapital stock. Because capital is below its steady state, however, investmentpicks up. Individuals consume less and eventually work more.By contrast, in the model with frictions in the intermediation process,

there is a sharp recession. The deterioration in asset quality produces amagnified decline in intermediary capital. The interest rate spread skyrocketsas a consequence, and output tanks. Output initially falls about three percentrelative to trend and then decreases to about six percent relative to trend.Though the model does not capture the details of the recession, it doesproduce an output decline of similar magnitude. Recovery of output to trenddoes not occur until roughly five years until after the shock. This slowrecovery is also in line with current projections. Contributing to the slowrecovery is the delayed movement of intermediary capital back to trend. It ismirrored in persistently above trend movement in the spread. Note that overthis period the intermediary sector is effectively deleveraging: It is buildingup equity relative to assets. Thus the model captures formally the informal

20

notion of how the need for financial institutions to deleverage can slow therecovery of the economy.We now consider credit interventions by the central bank. Figure 3 con-

siders several different intervention intensities. In the first case, the feedbackparameter ν in the policy rule given by equation (38) equals 50. At this value,the credit intervention is roughly of similar magnitude to what has occurredin proactive. The solid line portrays this case. In the second, the feedbackparameter is raised to 500, which increases the intensity of the response.The dashed line portrays this case. Finally, for comparison, the dashed anddotted line portrays the case with no credit market intervention.In each instance, the credit policy significantly moderates the contrac-

tion. The prime reason is that central intermediation dampens the rise inthe spread, which in turn dampens the investment decline. The moderateintervention (ν = 50) produces an increase in the central bank balance sheetequal to approximately ten percent of the value of the capital stock. Thisis roughly in accord with the degree of intervention that has occurred inpractice. The aggressive intervention further moderates the decline, thoughthe gain relative to the moderated intervention is small. In this case, centralbank lending increases to approximately twenty percent of the value of thecapital stock.

3.3 Optimal Policy and Welfare

We now consider the welfare gains from central bank credit policy and alsocompute the optimal degree of intervention. We take as the objective thehousehold’s utility function.We start with the crisis scenario of the previous section. We take as

given the Taylor rule for setting interest rates. This rule may be thought ofas describing monetary policy in normal times. We suppose that it is creditpolicy that adjusts to the crisis. We then ask what is the optimal choiceof the feedback parameter ν in the wake of the capital quality shock. Indoing the experiment, we take into account the efficiency costs of centralbank intermediation, as measured by the parameter τ . We consider a rangeof values for τ .Following Faia and Monacelli (2007), we begin by writing the household

utility function in recursive form:

Ωt = U(Ct, Lt) + βEtΩt+1 (39)

21

We then take a second order approximation of this function about the steadystate. We next take a second order approximation of the whole model aboutthe steady state and then use this approximation to express the objectiveas a second order function of the predetermined variables and shocks to thesystem. In doing this approximation, we take as given the policy-parameters,including the feedback credit policy parameter ν.We then search numericallyfor the value of ν that optimizes Ωt as a response to the capital quality shock.To compute the welfare gain from the optimal credit policy we also com-

pute the value of Ωt under no credit policy. We then take the difference inΩt in the two cases to find out how much welfare increases under the opti-mal credit policy. To convert to consumption equivalents, we ask how muchthe individuals consumption would have to increase each period in the nocredit policy case to be indifferent with the case under the optimal creditpolicy. Because we are just analyzing a single crisis and not an on-going se-quence, we simply cumulate to the present value of consumption-equivalentbenefits and normalize by one year’s steady state consumption. Put differ-ently, we suppose the economy is hit with a crisis and then ask what are theconsumption-equivalent benefits from credit policy in moderating this singleevent. Since we are analyzing a single event, it makes sense to us to cumu-late up the benefits instead of presenting them as an indefinite annuity flow,where most of the flow is received after the crisis is over.Figure 4 presents the results for a range of values of the steady state

markup and also a range of values of the efficiency cost τ . In the baselinecase of with a fifteen percent markup and no efficiency cost (τ = 0), thebenefit from credit policy of moderating the recession is worth 6.50 percent ofone years recession. At reasonable levels of the efficiency cosy (e.g. ten basispoints), the gain is on the order of 5.0 percent of steady state consumption.It decreases to zero, as the efficiency costs goes to forty basis points, as fairlylarge number. Though we do not report the results here, for τ less than fortybasis points the optimal credit policy comes closely to fully stabilizing themarkupThe welfare loss is increasing in the steady state distortion. If we interpret

this distortion from the markup as also capturing distortionary tax effects,then a much high value may be justified. For for example, average effectivelabor taxes are in the range of thirty to thirty five percent. Assuming theprice markup is in the neighborhood of fifteen to twenty percent, then asteady state distortion up to fifty percent is reasonable. Accordingly, weconsider two alternatives to our baseline: one involving a thirty-three percent

22

markup and the other a fifty percent markup. As the figure illustrates, thegains to the optimal credit policy increase substantially, increasing to overfourteen percent of one year’s consumption for the tau equals zero case andremaining above ten percent as tau reaches twenty basis points.One factor moderating the welfare loss is that the marginal disutility of

labor declines during the downturn. It could be that the simple preferencestructure we use to pin down labor supply gives a misleading read on theutility gains from increased leisure during recession. Accordingly, we redothe exercise, this time considering the case of perfectly elastic labor supply.In this case individuals only care about consumption fluctuations. As figure6 illustrates, in this case the benefits from the optimal credit policy increaseroughly twenty percent across the board.

4 Concluding Remarks

We developed a quantitative monetary DSGEmodel with financial intermedi-aries that face endogenously determined balance sheet constraints. We thenused the model to evaluate the effect of expanding central bank credit inter-mediation to combat a simulated financial crisis. We find that the welfarebenefits may be substantial if the efficiency costs of government interventionare modest.If we abstract from the issue of efficiency costs, an equivalent type of credit

intervention in our model would be direct equity injections into financialintermediaries. Expanding intermediaries equity would of course expand thevolume of assets that they can intermediate. In our view, a key factor inchoosing between these two policies involves the efficiency costs of the policyaction. For certain types of lending, e.g. securitized high grade assets such asmortgaged=backed securities, the costs of central bank intermediation mightbe relatively low. In this case, direct central bank intermediation may bejustified. In other cases, e.g. C&I loans that requires constant monitoringof borrowers, central bank intermediation may be highly inefficient. In thisinstance, capital injections may be the preferred route. By expanding ourmodel to allow for asset heterogeneity we can address this issue.Finally, we consider a one time crisis and evaluated the policy response.

In subsequent work we plan to model to the phenomenon as an infrequentlyoccurring rare disaster in spirit of Barro (2009) and others. In this literaturethe disaster is taking as a purely exogenous event. Within our framework we

23

can evaluate the gains from various policy responses, using the same tools asapplied in this literature to compute welfare.

24

References

References

[1] Adrian, Tobias, and Hyun Shin, 2009, "Money, Liquidity and MonetaryPolicy," mimeo.

[2] Aiyagari, Rao and Mark Gertler, 1990, "Overreaction of Asset Prices inGeneral Equilibrium," Review of Economic Dynamics

[3] Bernanke, Ben and Mark Gertler, 1989, "Agency Costs, Net Worth andBusiness Fluctuations," American Economic Review

[4] Bernanke, Ben, Mark Gertler, and Simon Gilchrist, 1999, "The FinancialAccelerator in a Quantitative Business Cycle Framework," Handbook ofMacroeconomics, John Taylor and Michael Woodford editors.

[5] Carlstrom, Charles and Timothy Fuerst, 1997, "Agency Costs, NetWorth and Business Fluctuations: A Computable General EquilibriumAnalysis", American Economic Review

[6] Christiano, Lawrence, Martin Eichenbaum and Charles Evans, 2005,"Nominal Rigidities and the Dynamics Effects of a Shock to MonetaryPolicy,", Journal of Political Economy

[7] Christiano, Lawrence, Roberto Motto and Massimo Rostagno, 2005,"The Great Depression and the Friedman Schwartz Hypothesis," Jour-nal of Money Credit and Banking

[8] Faia, Ester and Tommaso Monacelli, 2007, "Optimal Interest RateRules, Asset Prices and Credit Frictions," Journal of Economic Dy-namics and Control

[9] Geanakopolis, John and 2009, "Leverage Cycles and the Anxious Econ-omy," American Economic Review

[10] Gertler, Mark, Simon Gilchrist and Fabio Natalucci, 2007, "ExternalConstraint on Monetary Policy and the Financial Accelerator,’ Journalof Money, Credit and Banking.

25

[11] Gilchrist, Simon and Egon Zakresjek, 2009,

[12] Jermann, Urban and Vincenzo Quadrini, 2008, "Financial Innovationsand Macroeconomic Volatility," mimeo.

[13] Justiniano, Alejandro, Giorgio Primiceri, 2008, "Investment Shocks andBusiness Cycles," mimeo.

[14] Kiyotaki, Nobuhiro and John Moore, 2007, "Credit Cycles," Journal ofPolitical Economy

[15] Kiyotaki, Nobuhiro and John Moore, 2008, "Liquidity, Business Cyclesand Monetary Policy," mimeo.

[16] Lorenzoni, Guido, 2008, "Inefficient Credit Booms,"Review of EconomicStudies.,

[17] Mendoza, Enrique, 2008, "Sudden Stops, Financial Crises and Leverage:A Fisherian Deflation of Tobin’s Q,"

[18] Sargent, Thomas J. and Neil Wallace, "The Real Bills Doctrine versusthe Quantity Theory of Money," Journal of Political Economy

[19] Smets, Frank and RafWouters, 2007, "Shocks and Frictions in U.S. Busi-ness Cycles: A Bayesian DSGE Approach," American Economic Review

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[21] Woodford, Michael, 2003, Interest and Prices.

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Table 1: Parameter Values for Baseline ModelHouseholds

β 0.995 Discount rateh 0.700 Habit parameterχ 5.584 Relative utility weight of laborϕ 0.333 Inverse Frisch elasticity of labor supply

Financial Intermediariesλ 0.383 Fraction of capital that can be divertedξ 0.003 Proportional transfer to the entering bankersθ 0.972 Survival rate of the bankers

Intermediate good firmsα 0.330 Effective capital shareu 1.000 Steady state utilization rate

δ(u) 0.025 Steady state depreciation rateζ 1.000 Elasticity of marginal depreciation with respect to utilization rate

Capital Producing Firmsηi 2.500 Inverse elasticity of net investment to the price of capital

Retail firmsε 11.000 Elasticity of substitutionγ 0.750 Probability of keeping prices fixedγP 0.500 Measure of price indexation

Governmentκπ 1.500 Inflation coefficient of the Taylor ruleκX -0.500 Output gap coefficient of the Taylor ruleGY

0.200 Steady state proportion of government expenditures

27

Figure 1: Responses to Technology (a) , Monetary (m) and Wealth (w) Shocks

0 20 40−0.015

−0.01

−0.005

0Y

a

0 20 40−0.1

−0.05

0

0.05I

0 20 40−1

0

1

2x 10

−3 Rk−R

0 20 40−10

−5

0

5x 10

−3 Y

m

0 20 40−0.06

−0.04

−0.02

0

0.02I

0 20 40−5

0

5

10

15x 10

−4 Rk−R

0 20 40−4

−2

0

2x 10

−3 Y

N

0 20 40−0.02

−0.01

0

0.01I

0 20 40−5

0

5

10x 10

−4 Rk−R

FA SDGE

1

Figure 2: Responses to a Capital Quality Shock

0 20 40

−0.04

−0.02

0s

0 20 40−5

0

5x 10

−3 R

0 20 40−0.02

0

0.02Rk−R

0 20 40−0.1

0

0.1Y

0 20 40

−0.04

−0.02

0

C

0 20 40−0.5

0

0.5I

0 20 40−0.2

−0.1

0K

0 20 40−0.05

0

0.05L

0 20 40−0.2

0

0.2Q

0 20 40−1

−0.5

0N

0 20 40−2

0

2x 10

−3 π

0 20 40−5

0

5x 10

−3 i

FA SDGE

2

Figure 3: Responses to a Capital Quality Shock with Credit Policy

0 20 40

−0.04

−0.02

0s

0 20 40−5

0

5x 10

−3 R

0 20 40−0.02

0

0.02Rk−R

0 20 40−0.1

0

0.1Y

0 20 40

−0.04−0.02

0

C

0 20 40−0.5

0

0.5I

0 20 40−0.2

−0.1

0K

0 20 40−0.05

0

0.05L

0 20 40−0.2

0

0.2Q

0 20 40−1

−0.5

0N

0 20 40−2

0

2x 10

−3 π

0 20 40−5

0

5x 10

−3 i

0 20 400

0.2

0.4ψ

CP CP ν=500 CP ν=0

3

Figure 4: One year consumption equivalent welfare gains from optimal credit policy as afunction of efficiency costs tau and steady state markup

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

τ

Ω

CP X=15% CP X=33% CP X=50%

4

Figure 5: One year consumption equivalent welfare gains from optimal credit policy as afunction of efficiency costs tau and steady state markup with high labor supply elasticity

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

τ

Ω

CP X=15% CP X=33% CP X=50%

5

Figure 6: Credit vs. premium policy

0 20 40

−0.04

−0.02

0s

0 20 40−5

0

5x 10

−3 R

0 20 40−0.02

0

0.02Rk−R

0 20 40−0.1

0

0.1Y

0 20 40

−0.04

−0.02

0

C

0 20 40−0.5

0

0.5I

0 20 40−0.2

−0.1

0K

0 20 40−0.05

0

0.05

L

0 20 40−0.2

0

0.2Q

0 20 40−1

−0.5

0N

0 20 40−2

0

2x 10

−3 π

0 20 40−5

0

5x 10

−3 i

Premium policy Credit policy

6